In using a slide wire potentiometer, show a proof for the equation Vx = Cx where V = the voltage drop, and C = the position along the slide wire potentiometer.
To understand the proof for the equation Vx = Cx for a slide wire potentiometer, let's start by understanding the basic setup of a slide wire potentiometer.
A slide wire potentiometer consists of a long wire with a constant resistance along its length, with one end connected to a voltage source. A sliding contact, often in the form of a movable wiper, is connected to a voltmeter. By moving the wiper along the wire, we can measure different voltage values at different positions.
Now let's go through the steps to prove the equation Vx = Cx:
1. The total resistance of the slide wire potentiometer is constant since the resistance per unit length is constant. Let's call this constant resistance R.
2. When the sliding contact (wiper) is at a position C (normalized between 0 and 1, where 0 represents one end and 1 represents the other end), the resistance between the wiper and the end connected to the voltage source is R*C (as C represents the fraction of the total length of the potentiometer wire).
3. According to Ohm's Law, the voltage drop (V) across a resistor can be calculated as V = I * R, where I is the current flowing through the resistor.
4. In the slide wire potentiometer, the current flowing through the wire is the same throughout its length, as there are no branches or junctions.
5. Therefore, the voltage drop (Vx) between the sliding contact (wiper) and the fixed end connected to the voltage source can be calculated as Vx = I * (R * C).
Now, we have Vx = I * (R * C).
To further simplify, let's assume the current (I) is constant, and we can divide both sides of the equation by I:
Vx / I = R * C.
Since Vx / I represents the voltage drop across the resistance (Vx) and R * C represents the resistance between the sliding contact (wiper) and the fixed end connected to the voltage source, we can state:
Voltage drop (Vx) = Resistance (R * C).
Therefore, we have successfully proved the equation Vx = Cx, where Vx represents the voltage drop and C represents the position along the slide wire potentiometer.